Chapter 2. Conditional Statements

Irving M. Copi, Symbolic Logic, New York: Macmillan Publishing, 1979, pp. 16-18.

 

(2) Conditional Statements

① The compound statement "If the train is late, then we shall miss our connection" is a conditional (or a hypothetical, an implication, or an implicative statement).

② The component between the 'if' and the 'then' is called the antecedent (or the implicans or protasis조건문의 조건절, 전제절), and the component that follows the 'then' is the consequent (or the implicate or apodosis조건문의 귀결절).

③ A conditional does not state either that its antecedent is true or that its consequent is true; it says only that if its antecedent is true, then its consequent is true also, that is, its antecedent implies its consequent. The key to the meaning of a conditional is the relation of implication that is asserted to hold between its antecedent and its consequent, in that order.

④ A common partial meaning of these different kinds of conditional statements emerges when we ask what circumstances would suffice to establish the falsehood of a conditional. Under what circumstances would we agree that the conditional "If this piece of gold is placed in this solution, then this piece of gold dissolves" is false? Clearly, the statement is false in the case that this piece of gold is actually placed in this solution and does not dissolve. Any conditional with a true antecedent and a false consequent must be false. Hence any conditional if p, then q is known to be false in case the conjunction p ∧ ~q known to be true, that is, in case the antecedent of the conditional is true and its consequent false. For the conditional to be true, the indicated conjunction must be false, which means that the negation of that conjunction must be true. In other words, for any conditional if p, then q to be true, ~(p ∧ ~q), the negation of the conjunction of its antecedent with the negation of its consequent, must be true also. We may, then, regard the latter as a part of the meaning of the former.

⑤ We introduce the new symbol '⊃,' called a horseshoe, to represent the partial meaning that is common to all conditional statements. We define 'p ⊃ q' as an abbreviation for ~(p ∧ ~q). The horseshoe is a truth-functional connective, whose exact significance is indicated by the following truth table:

Here, the first two columns represent all possible truth values for the component statements p and q, and the third, fourth, and fifth represent successive stages in determining the truth value of the compound statement ~(p ∧ ~q) in each case. The sixth column is identically the same as the fifth, since the formulas that head them are defined to express the same proposition. The horseshoe symbol must not be thought of as representing the meaning of 'if-then,' or the relation of implication, but rather symbolizes a common partial factor of the various different kinds of implications signified by the 'if-then' phrase.

⑥ The weak implication symbolized by '⊃' is called a material implication. Its special name indicates that it is a special concept and not to be confused with the other, more usual kinds of implication. Although most conditional statements express more than a merely material implication between antecedent and consequent, we now propose to symbolize any occurrence of 'if-then' by the truth-functional connective '⊃.' Such symbolizing abstracts from or ignores part of the meaning of most conditional statements. But the proposal can be justified on the grounds that the validity of valid arguments involving conditionals is preserved when the conditionals are regarded as expressing material implications only, as will be established in the following section.

⑦ Conditional statements can be expressed in a variety of ways.

any of these formulations will be symbolized as p ⊃ q	if p, then q
	if p, q
	q if p
	that p implies that q
	that p entails that q
	p only if q
	that p is a sufficient condition that q
	that q is a necessary condition that p

 

Exercises

1. A ∧ C .⊃. ~D
2. A .⊃. C ∨ D
3. A .⊃. C ∧ D
4. A .⊃. C ∨ D
5. ~A ⊃ ~(C ∨ D)
6. ~(A ∧ C) ⊃ (C ∧ D)
7. A ⊃ ~(C ∧ D)
8.~A .⊃. ~C ∧ ~D
9. A ∧ ~C .∨. C ⊃ ~D
10. A ⊃ ~C .∧. ~C ⊃ D
11. A .⊃. ~C ⊃ D
12. A ∧ C .∨. ~(C ⊃ D)
13. A .⊃. ~C ∨ ~D
14. A ⊃ C .⊃.  ~D
15.  ~A ∧ ~C ⊃ ~(A ∧ C)